Algebraic polynomial generator



Feb. 25, 19 58 F. H. RAYMOND 2,324,690

ALGEBRAIC POLYNOMINAL GENERATOR Filed July 17, 1952 4 Sheets-Sheet 1 ji' NEGATIVE 0mm AMPLIFIER 2 I //6 INVENTOR.

+Y4 FRANCOIS HENRI RAYMOND III -Y4 imam BY ,(w

AT roam-77 F. H. RAYMOND Feb. 25, 1958 ALGEBRAIC IEOLYNOMINAL GENERATOR 4 Sheps-Sheet 3 Filed July 17, 1952 D.C.AMPL|FIER INVENTOR. ."Y FRANCOIS HENRI RAYMOND uMMnvq MPLIFIER AT TORNEY are derived:

United States Patent 2,824,690 ALGEBRAIC POLYNOMIAL GENERATOR Franeois Henri Raymond, Saint-Germain-en-Laye, France,

assignor to Societe dElectroniqueet dAutomatisrne,'

Courbevoie, France Application July 17, 1952, Serial No. 299,436 Claims priority, application, France July 26, 1951 40 Claims. (Cl. 235-61) The present invention concerns improvements related to the realization of electric calculators designated to handle algebraic polynomials of nth degree of the general form:

Variable z is a complex variable such as:

portions x and of the imaginary portions y; of the complex variable, represented by the electric control voltages of the calculator, are varied inthe operator, manually or in accordance with a predetermined automatic or semiautomatic program. a

' By regrouping the terms of even and odd powers of 2:, expression (1) can also be written:

With substitution of 2+iy2. z*= 4+iyr.

expression (3) becomes:

-ls- 4+ 7- e+ In designating:

the expressions for the real and imaginary portions of the'polynomial P(z) become:

=23 +x .24+y .2 2 From relations (2) and (4), on the other hand, there and so OIT'CIOSCI' and closer, for the power yaluesof 2 because z =z-.z and generally, for It even there is:'

n 2- n-2 2- n2 and u= n-a- 2+ a-Js-a These and other objects of the invention will be more fully described in the drawings annexed herewith for the case of an algebraic equation of the 7th degree.

However, extension or restriction to any higher or lower degrees follows directly from the diagrams given.

The invention will be further explained, again for illustration only, as applied to a calculator for the determination of the real and imaginary roots of equation P('z)=0. Numerous other applications can be envisaged once the voltages representing R(z) and I (z), have bee formed, respectively. I

In the drawings, Figs. 1 and 2, together, Fig. 1 being left of Fig. 2,

show an example of a diagram of a calculator according to the invention.

Fig. 3 which can be substituted for Fig. 1 in this assembly shows a variation of this embodiment.

Fig. 4 presents an example of using an element in the diagram of the calculator of Fig. 3 and- Fig. 5 represents a control diagram for the indicator to determine the real and imaginary roots of an algebraic equation derived from the voltages R(z) and I (z) produced for the calculator of Figs. 1-2 or 3-2.

Referring in the first place toFigs. l and 2, the two variables x, and y; are introduced in' a calculator over shafts 1 and 23. These shafts can be operated or put into position manually and/or automatically by any desired means (not shown).

The control of these shafts will be described in greater detail further below, as applied in the calculator to the determination of the roots of an algebraic equation.

Shaft 1 if rotated drives potentiometers sliders 2' to 6. The potentiometers themselves are designated as 7 to 11 respectively. Potentiometer 7 receives at its terminals a fixed continuous potential difference or reference potential +V, V. Shaft l-is of one piece or rigidly coupled with the rotor of motor 12 with a separate excitation winding, for example, indicated at 13 and supplied from amplier 14. Input of amplier 14 is operated upon from a mixer with two resistances 16 and 17. Resistance 16 is connected over lead 15 to slider 2 of potentiometer 7. Resistance 17 is connected to a take 011 terminal of voltage x Such voltage if applied to terminal x, of the servomechanism to control the angular position of shaft 1. 1

The positioning servomechanism referred to is of the usual type but if necessary can be made inverse functioning it the given element is the position of slider 2 on potentiometer 7, which determines electric voltages x while in a servomechanism in habitual application, there is the applied voltage x that will determine the position of the slider in equilibrium.

It should be well understood that all the otentiometers drawn in linear shape are in fact of circular structure. The sliders are in the form of brushes driven by the mechanical axes shown.

Voltage +x taken off by means of connection 18 is applied at one end of potentiometer 8. The other end of potentiometer 8 receives voltage --x;; the latter is produced by polarity inversion in amplifier stage 19 of gain (-1). This stage consists as usual of an amplifier tube 19 of elevated gain operated by resistance mixer 20-21; resistance 20 receives electric voltage +x the output voltage of stage 19 is taken over resistance 21; all resistances have the same value as the resistances of the other mixers shown in the diagrams.

Product .of .the voltage at terminals with .th indication coefiicient (principle of the multiplying potentiometer).

At take off connection 43 of slider 3, therefore, there will appear a voltage proportional to x Shaft 23 if rotated, drives sliders or brushes 24 to 27 of po'ten'tiome'tersis to 31, respectively; potentiometer 28 receive's at its terminals the same reference voltage :Vas potentiometer 7. .Shaft23 is driven by motor 32, separately .excited at 33, of a servomechanism assembly: the voltage applied to slider 24 by conductor 35 is;taken off resistance 36 of input mixer 36-37 of amplifier stage -34 feeding coil33.

'At the'freeendnf resistance 37, there appears :anelec- :tric voltage :rcpresenting variable y available :at the :terminalofEthesarne.reference. Through connection 33, ahis voltage +ry ;is applied onone-end of potentiometer 29; at the other end of potentiometer 29,-voltage ---y is applied. Voltage y is produced .bytthe polarityinversionrstage of ;unity gain of amplifier tube .39 jand mixer-4H1. ,-Resistance'40r eceives voltage and resistance 41 the output voltage of stage 39. I

EHere againithe voltagenaken .otf .51id l'::1 '.8fld appe ing on connection 45,:represents the Product of the-volt- :age -.at.- terminals of ..-poten tiometer 29 indicating the position of slider 25, hence -y ;,not e that the senses jof conneefioris for otentiometers 4 Hand ;29 are reversed. Conductor 43 is connected to-resistance and con- .ductor 45 ,to resistance46 of the input mixerof amplifier stage 47. The output of stage 47 supplies excitation coil of:motor-=49-drivingshaft '50. The yoltage taken off ,slid,er -51 of potentiometer .56 is -,taken over-connection ;61 qnto ,the-third input resistance 62;of ihe mixer. Slider 5 1 is.rigidlycoupled with shaft- 50 ofnnotor .49 and potentiometer 56 receives at its terminals the reference voltage. This assembly constitutes therefore the position servomechanism andthe-voltagetaken off the slider is the algebraic sum ofthe input-voltages. Hence-in this case the .voltage is -x =x 'y Thisis 'the voltage ;applied over resistance-63-on the input of-inversion stage sat-gain (-1 The tfl itput voltage ofvstage 64 is .talcen over resistance 65 to the input-mixer. At terminals 66 and 67, therefore, there appears an .electric voltage representing variable +13. On the other hand, since the =servomechanisrn is of the position typeyvariable v+x is equally available onvshaft 50 in the form of an angle of rotation ormore precisely of anangular position. The 'VflIUOOf: variable :5 thus producedisthereafterindicated pn otentiometers .57 :to .-60 by the.e positions of sliders 219.55, 4

;P,otentio'rnetcr .57. ver connec ion 72- vreceives at'one end the voltage +x at its other end iii-receives voltage ---,x= produced by polarity inversion stage :69 of gain (l). Stage .69-delivers .this voltage to points 71 and 73; it has an input mixer 68-30; resistance 68 is connected to output67 of stage 64; resistance 70 feeds the output voltage taken off at 73 back to the input. On conductor 100 of slider 52, ;therefore, there appears a voltage representing the value x}.

Potentiometer-9 coupled to shaft 1 over slider 4, receivesat its terminals over connections 42 and 38 the voltages -y and-+J'1- The voltage 'taken off slider 4 therefore is proportional to the product x y (x is the coefficient indicated by its slider).

This voltage is taken over connection 7410 the input .of resistance 75 of the-input mixer of a servomechanism to produce mechanically and electrically representations 'Qf-HURMIIYQFZXQ' This servomechanism'is a position :mechanisrn'of the .usual type. =It comprises =amplifier76 which fe'cdsertcitation 77 .of-motor 78-driving shaft 79; -the angular :position of the latter furnishes y, in-meplie'd ieference -potential ditfegence 'i'V; the connection -of lslider fiiir is led-back to input-resistance S8 of the input 'mix'er of the amplifier. At slider 80, therefore, there is available the electric voltage representing This voltage is first inversed by amplifier stage of gain (1). Stage 90 receives this voltage at input resistance 89. Its other input resistance 91 receives the voltage fed back from its output. Electric voltage y, therefore is available at 92 and 93. This voltage is again inversed in amplifier stage 95 of gain (l). .It is applied .over input resistor 94 while input resistance 96 receives the voltage fed back from the output ofamplifier'95. Voltage y; is therefore available at 95 and-97.

Potentiometer 85 receives at its terminals potential difference i'yg; since the position of its slider indicates the value y the voltage taken off conductor 102 of slider 81 will -represent the square y}. it .will'have sign with respect to the voltage representing +x The latter is taken off at 100 by inversion relative to the connection senses of potentiometers 57 and 85.

Conductors 100 and 102 are connected respectively to 'resistances, 101-and '103 of input mixer of summingamplifier-stage 104. The output-voltage of stage 104 isfed back to the resistance 105 of that mixer in such -a way that a voltage representing is available at terminals 105 and 107.

This output voltage +x 'is then polarity inversed in polarity in stage 109 of gain (l). Stage 109 receives this voltage over input resistance 108. The other input resistance 111 is connected to output upon which there :appearsfias zwelL-as on .output .112.-+the electric voltage -x Potentiometer 58 over-conductors 98 and receives potential difference In Its slider 53 indicates coefficient x :The voltage taken off its slider overconductor 113, therefore, represents electric voltage -y This voltage fed over input resistance 114 of stage --115 of gain '(-1)--the other input'resistance 117 receiving the returning output voltage-appears in inverse polarity at terminals :116 and '118. Moreover, this :voltage +y applied over input resistance 119 is again inversed in stage 120 of gain (-1). The other resistance 122 receives the voltage fed a back from output 121. This establishes a voltage --y available at terminals 121 and 123.

:For the production ofelectric .-voltages representing values x and y circuits are now established in such a manner as to realize relation (9). In order to achieve this purpose,-.terminals 107,.112of Fig. 1 brought to the votlages +x x respectively, are connected at 124, 125 to the terminals of potentiometer 59. Slider 54 indicates value x:; thus the voltage taken off thatslider over conductor is proportional to product x 41 Terminals 118-123 of Fig. 1 brought to voltages +3! and y respectively, are fed over conductors 128, 129 to the terminals of the potentiometer 86. The slider 82 of potentiometer 86 indicates value y From that slider therefore over conductor 132 voltage -y .y is derived. Sign is to relate to the voltage taken off conductor 130 due to the relative inversion of the connection senses of the connection of these potentiometers.

The two voltages are added in summing amplifier 134, being applied-on resistances 131 and 133 respectively of the input mixer. The third resistance 136 of that input mixer receives the returning output voltage at 135. This output voltage here is +x taking into account the affect of the sign of the preceding outputs of Fig. 1.

Output voltage +x, is furthermore applied over input resistance'137 to inversion stage 138 of gain (l). Output 139 of stage 138 is fed back over the other resistance 1140 of the inpnLmixer. gAt 139,-therefore, voltage x, is available.

Similarly, over-conductors 128 and 129 potentiometer jo-receives at itsends the potentialrdiflereneef-gy while potentiometer 87 receives-.at its-.endsthe potential difference ix .;S lider ;55 .pf ;potentiometer;;60 indicates value x, and slider 83 of potentiometer 87 indicates value 3' thus the-output voltages taken on 'thesesliders over wires 141 and 145, represent with the same sign the factors x .y and Xg-Jz. Addition to. these voltages, respectively, applied on resistances 142 and 146 of the input mixer of summing amplifier 143-thethird resistance 147 of which receives the voltage fed back from output 144-delivers therefore voltage y This voltage is inversed in stage 149 of gain (1), being applied over resistance 148 on the input mixer of that stage. The other resistance 151 is connected to output 150. At output 150 of that ampifier stage, therefore, there appears voltage +y The chosen example stops at values 2x and iy It is apparent, however, that from these two voltage values voltages :x and iy can be produced. Thereafter, from these latter values, the two voltage values ix and iy can be derived and so forth in cascade development.

It will sufiice for that purpose to couple shafts 50 and 79 of as many assemblies of four potentiometers similar to potentiometers 59, 60 and 8687, and to connect their terminals to summing outputs delivering voltages of even index immediately inferior, and to connect sliders to summing inputs similar to stages 134 to 149.

It is noted that the preceding production of voltage values -x and i3 reverts to the latter method because in the particularization of relation (9) in order to pass from ix iy to :x.;, in n had to be 4.

The elected voltages representing algebraic sums E2 and 24 defined above are formed, Fig. 2, by means of amplifier summing stages 157 and 166, respectively. The component voltages are applied individually over the input resistances of mixers 156 and 165 of stages 157 and 166;

The input mixer of stage 157 receives the following voltages:

From the slider of potentiometer 152 where there is applied reference voltage :V, stage 157 receives a voltage proportional to a From the slider of potentiometer 153 where there is applied the voltage ix stage 157 receives a voltage proportional to product a .x the value of coeflicient a;, of the algebraic equation is indicated by the position of that slider.

From the slider of potentiometer 154 where there is applied the voltage in, stage 157 receives a voltage proportional to product (1 the value of coefficient a of the algebraic equation is indicated by the position of that slider.

From the slider of potentiometer 155 where there is applied the voltage i-x stage 157 receives a voltage proportional to the product mor the value of coefficient a-, of the algebraic equation is indicated by the position of that slider.

The input mixer 165 of the summing stage 166 receives the following voltages:

From the slider of potentiometer 162 where there is applied voltage iy stage 166 receives a voltage proportional to product a .y the value of coeflicient a of the algebraic equation is indicated by the position of that slider.

From the slider of potentiometer 163 where there is applied the voltage -y the stage 166 receives a voltage proportional to the product (1 1 value of coeiiicient a of the algebraic equation as indicated by the position of that slider.

From the slider of potentiometer 164, where there is applied the voltage iy stage 166 receives a voltage proportional to product a .y the coefficient a of the algebraic equation has its value indicated by that slider.

Electric voltage 22 must be multiplied by the value of variable x and by that of variable y In order to achieve this purpose, in the first place, the voltage available at the output of summing stage 157 is applied over conductors 161 and 174 on a terminal marked of potentiometers 10 and 31., Slider 5 of potentiometer 10 is coupled to shaft 1 which isthe shaft of the x and slider 27 of potentiometer 31 is coupled to shaft 23 which is the shaft of the 31;. On the other hand, this voltage is inversed in amplifier stage 159 of gain (l) with input mixer 15%. The voltage of inverse polarity is applied over conductors 160 and 173 on the terminals marked of the same potentiometers 10 and 31.

Electric voltage 24 must be multiplied by the value of variable x and by that of variable y In order to achieve this purpose, in the first place, the voltage available at the output of summing stage 166 is .applied over conductors 169 and 172 on terminals marked of potentiometers 38 and 11. Potentiometer 30 has its slider 26 coupled to shaft 23 which is the shaft of the y On the other hand, this output voltage of stage 166 is inversed in amplifier stage 163 of gain with input mixer 167. The voltage of inverse polarity is applied over conductors 171 and on the terminals marked of the same otentiometers.

In this way the following voltages are available:

Voltage x 22 on conductor 175 of slider 5 Voltage y .24 on conductor 176 of slider 26 Voltage x 24 on conductor 184 of slider 6 Voltage 3 L122 on conductor 135 of slider 27 Conductors 175 and 176 are connected to the two input resistances of mixer 177 of summing amplifier 178. The other input resistances receive the component voltages of the algebraic sum defined above 21, i. e.:

From the slider of potentiometer 179 receiving reference voltage iV, stage 178 receives a'voltage proportional to coefiicient a of the algebraic equation the value of which is indicated by said slider;

From the slider of potentiometer 180, receiving the voltage 1x stage 178 receives a voltage proportional to product a .x the value of coefficient a; of the algebraic equation is indicated by the position of said slider;

From the slider of potentiometer 131 receiving voltage in, stage 178 receives a voltage proportional to product 0 x the value of coefiicient a of the algebraic equation is indicated by the position of that slider;

From the slider of potentiometer 182. receiving the voltage ix stage 178 receives a tension proportional to product a .x the value of the coefficient a of the algebraic equation is indicated by the position of that, slider.

Mixture and addition of these different voltages assure, therefore, the production at output 183 of summing amplifier 178 of an electric voltage representing the real portion R(z) of the algebraic equation in conformance with relation (6).

Conductors 184 and 185 are connected to the two input resistances of mixer 136 of a summing amplifier 187. The other resistances of stage 187 receive the component voltages of the algebraic sum above defined E3, 1. e.:

From the slider of potentiometer 188 receiving the voltageiy stage 187 receives a voltage proportional to product a .y the value a of the coefiicient of the algebraic equation is indicated by the position of that slider;

From the slider of potentiometer 189 receiving the voltage in, stage 187 receives a voltage proportional to product 11 0 the value of coefiicient a of the algebraic equation is indicated by the position of that slider;

From the slider of potentiometer 1% receiving the voltage iy stage 137 receives a voltage proportional to product a .y thevalue of coefficient a of the algebraic equation is indicated by the position of that slider.

Mixture and addition of these different voltages, therefore, assure the production at output 191 of summing amplifier 187' of an electric voltage representing the imaginary portion 1(2) of the algebraic equation in conformance \vithrelation (7). w Fig. 3 shows a variation of the realization of Fig.1.

utilizing mechanical calculator for the produetion of variable quantities -x y and x -y This implies -the following equations:

Shaft 192 for indicating the values of a controls mechanism 193-194 as illustrated in greater detail in Fig. 4 which assures driving of shaft 197 in accordance with the rotative displacement proportional to sin a=x Shaft 197 is ,of one piece rigidly coupled with shaft 1 mentioned above.

By means of a reduction gear mechanism, mechanism 193 drives mechanism 195 which causes shaft 196 to turn proportionally to quantity 1cos 24, i. e. 2x

Shaft 205 for indicating b controls mechanism 203-204 which assures driving of shaft 23 mentioned above in accordance with a rotative displacement proportioual to sin b=y By means of a reduction gear transmission, 203-204 drives mechanism 202 which drives shaft 200 with inversion of direction at 201 to cause itto turn proportionally to quantity 1cos 2b, i. 6. 23

Differential 197 controlled by the two shafts 196-and 200, drive shaft 198 which turns therefore with a rotation proportional to 2x .-2y If divided by 2 through transmission 199, this rotation represents the value of variable x and is therefore applied to drive shaft 50 above mentioned.

On the other side, shaft 206 of one piece .with or rigidly coupled to shaft 192, and shaft 207 solidly coupled with shaft 205 operate upon differentials 208 and {209; shaft 207 operates the latter differential 209 after inversion of direction of rotation at 210. Shaft 212 therefore ,turns proportional to (a+b) and shaft 211 proportional to (a-b). These two shafts drive mechanisms 21,4,and 213 respectively which resemble preceding mechanism 193, 204, i i at rotations which are proportional to quantities cos (a-b) and cos (a-l-b). 'With difierentiall217 operated upon by shafts 216 and 21S, therefore, shaft 79 mentioned above is drivenin conformance with variable y An example illustrating the mechanism for transforming a rotation proportional to a quantity, into a rotation proportional to a trigonometric line, sine or cosine of that quantity, or of a multiple of that quantity, is indicated in Fig 4. Two discs 219 and 221:are supported on shafts 218 and 220, respectively, shaft2 18 being as sumed to be driving. Discs 219 and 221 have the same radius and are connected by crossbar 22 2mounted eccentrically on each of the discs 219 and 221. Bar 222 supports a block 223 sliding between the guide rails 2 24 and225. Block 223 is solidly attached to belt 200 passing over rollers 227 and 229 supported on shafts 22 6 and 228, respectively. Shaft 22% is assumed to be the driven shaft. Block 223, of course, is free to slide over bar 222. Since shaft 218 is driven, bar 222 remains horizontal but its height will vary, for example, conforming to the variation of the trigonometric line, sineor cosin e, selected in accordance with the initial fixation of theeccentric supporting bar 222. By driving roller 229 through belt 230, this amplitude variation is translated into a variation of angular position.

The calculator just described in one or the other of its embodiments, therefore, permits to produce .at any instance the values of the real and im aginary portions of an algebraic equation of the type defined forthevalues controlled by variablesjt; and Illelatterdefi ne togetherthe value of variablelz of that equation. I

"T reli nc i alue o ari le n n v1 sai h achi v d indepe de y f ptnsac o he .o iim sc anc with any law of coordination preestablishfidrortimposed ath npstatq I il tt i ent r ly westernsin As a result shafts 216 and 215 are driven versely) eating .the multiplicity of values of :the equation coefficients'on :theindicationpotentiometers of the machine.

Considering for example that it is .desired gto resolve the-algebraiccquation i. eotoldetermine its real and'imaginary roots which means requiring the following determinations:

618 very srrralliandlarbitrarily determined; this determination maveasily .beconducted ,byztaking output termirials 183 and 191 of Fig. 2; they deliver electric voltages proportionaliuquantity and sense ;to the real and imaginary portions eR(z) .and 1(1) ofvthe equation and are connected .to .the inputs ,of the same references of the circuit diagrammatically represented in Fig. 5.

Ifhis circuit ,comprisesiin the first ,place, c onvent ional amplifiers 231 and 232 the output voltages ,ofwhich are rectified, .for example, hyfilry rectifier bridges 233 and 23.4. "The positive output voltages of these bridges are applied on=twolinput-resistancesof summingamplifier 235 of gain (.l'). Thereby, appearance at its output of a negative voltage is caused, indicating :the modul of Z if x and y arecaused to vary .in accordance with any predeterminedflavi.

Thislnegative voltage isappliedas blocking voltage on thesuppressor grid ofia three-grid tube 236. ,Tubej236 has at its suppressor grid permanently a positive voltage. At 243 there is indicated a voltage of PPOSllZiVfi polarization .of slight .value. This 1slight voltage will tend ,to hold tube =236 permanently conducting, were .it not for the negative voltagereceived fromstage 235. .Itis clear that each .time .the output .voltage.of vthat stage will ,become sufficiently small so that the suppressor gridPotential attainsx ground. potential 236 --,will be tde qckfi value of polarization 243 defines the amount of approximatiomof'theroots.

Tube 236.011 becoming conducting, ideliversza negative voltage which actuates a monostahle flip :fiop stage 237 byiblocking :the tube deblQckedinzthe rest positionpfthat stage. Otherwise, this stage .is ofusual construction and of a restoration time'deterrnined arbitrarily for a desired durationofoutput voltage pulse. 'Ihe o ipl .Of'fi P P stage 237, therefore, in positive, as soon as actuated operates ,upon' the control gridor Wehuelt .electrodeof a cathode ray tube indicator. The vertical and horizontal deflection coils 241 and 242 of the tube receive voltages ,representingtvariables x and y .As a resultas soon as the quantity of one or the other of these two variables is varied, the position of thefiatho ic spoton-screeuflti oftube 239will-vary accordingly. Shafts 9: and y are indicated upon screen 240. ,However, the illumination of the spot will tonly ccur if wehneltclectrode 238 receives a positive voltage from monostablefiip flop tstage 1237,.i. e. as soon as .the-modul ofZ passesthrqugh zero, within the desired approximation.

It is advantageousito provide:means ;(not shown) in front of screen ,240to assure :registratiou on a photographic platc, in the course of a determinationprocedure. The latter can.:,then be;condu ct ed in the following manner:

After having tindicatedi he .cqeffici utsiof t e qu t rtozbeisolvednthe operatormanqe vers the control shaft of thex; leaving at rest the controlshaft of the y (or con- If in this operation the value of modul Z passes through zero, the cathodic spot illuminates the :scmen uth Qshaft t 1an ;r1 w h i xp d-a reg trat o tofiha r o alu ak p h operator .thc ish t on s el tren rorder to .d pl ee thelia ru xn er t c zal us which he a rie -thcre j en heoths e uh tatiable i a s ntime .fa hion- I this way, henwil e wi s ub ai me D se eral m iliehe t kes back t w th va ab e ef-e plot tieu ;te h.i ith alu -vh sh s els zt m na ion o th l n abou t be explor d, n L f rth until the complex plane defined on screen 240 of oscillograph 239 is completely explored.

It is well understood that this exploration can be rendered automatic or semi-automatic, by providing an automatic or semi-automatic drive for shafts 1 and 23, Fig. 1, and shafts 192 and 205, Fig. 3.

Still further variations can be applied to the dispositions described without exceeding the scope of the invention such as defined by the preceding description.

The resulting calculators which deliver voltages representing real and imaginary portions of an algebraic equation are not limited to be used for the simple solution of equations such as have been just described in the procedure for root determination. Such calculators can also be incorporated in all types of chains for calculation or transmission of controlled quantities, all this without departing from the spirit of the invention.

: Iclaim:

1. In a system for computing the real and imaginary portions of a polynomial P(z) of nth degree, two pairs of adjustment means, the first pair of adjustments means being adjustable to positions corresponding to a pair of substitute variables representing the real and imaginary portions of 2 respectively, sources'of reference potentials controlled by said first pair of adjustment means to reproduce potentials representing respectively said pair of substitute variables, means including sources of potentials representing said substitute variables and controlled by said first pair of adjustments means to produce a first pair of derivative variables and control said second pair of adjustment means, a number of groups of further sources of potentials corresponding to a number of other pairs of derivative variables, each particular pair of said other derivative variables being derived from de: rivative variables preceding said particular 'pair of derivative variables; each group of said further sources including two sources of potentials representing a preceding pair of derivative variables and controlled by said second pair of adjustment means to produce potentials representing a succeeding pair of derivative variables; a first pair of groups of potential multiplying means including a number of sources of potentials representing all said pairs of derivative variables and adjustment means adjusted to produce two series of product potentials of said derivative variables and the even coefficients of P(z), and a second pair of groups of potential multiplying meansincluding a number of sources of potentials representing all said pairs of derivative variables and adjusted to produce two further series of product potentials of said derivative variables and the odd coetficients of P(z), means for adding said further two series of product potentials respectively, to produce two output potentials representing the sums of said of each of said further two series of product potentials said output potentials derived from said odd coefiicients being controlled by said first pair of adjustment means to produce four product potentials of sums of said second further pair of series of products multiplied by pairs of substitute variables; and

two final means eachfor adding two 'of the last-mentioned product potentials derived from said last pairs of sources and one of the two series of product potentials derived from said even coefiicients, to produce two output potentials representing respectively the real and imaginary portions offlz).

2." System "according-to claim 1 wherein at least some of said sources of potentials are in the form of multiplying otentiometers having sliders and at leastsome of said adjustment means include shafts coupled to said sliders.

3. Systemaccordingto claim 1 wherein atleast some of said adding means are in the form of amplifiers having several input circuits for summing potentials appliedv l 10 responsive servo-mechanisms controlling the angular movement of said shafts.

5. System' according to claim 1 comprising polarity inversion means coupled to at least some of said sources of potentials.

6. System according to claim 1 comprising polarity inversion means coupled to at least some of said adding means.

7. System according to claim 1 comprising polarity inversion means coupled to at least some of said sources of potentials, and polarity inversion means coupled to at least some of said adding means.

8. System according to claim 1 wherein said substitute variables are inserted in the form of z=x +jy 9. System according to claim 1 wherein said derivative variables are inserted in a sequence of alternating numerical order.

10. System according to claim 1 wherein said derivative variables are inserted in a sequence of even numerical order.

11. System according to claim 1.wherein said substitute variables are inserted in the form of z=x +jy and wherein said derivative variables are inserted in a sequence of even numerical order.

12. System according to claim 1 wherein said algebraic polynominal is of nth degree with real numerical coefficients and the complex variable is substituted by z =x +jy and is written directly:

there being defined: 7

E1, the algebraic sum (a +a .x +a .x +a .x 22, the algebraic sum (a -j-a .x +a .x =a .x 23, the alegbraic sum (a .y +a .y +a .y

E4, the alegbraic sum (a .y +a .y +a .y

and first derivative variables x and y being related to substitute variables x and y by relations:

and other derivative variables of higher orders being related to said first derivative variables x and y and pre ceding derivative variables x,, and y,, by relations (n being even): 7

13. System according to claim 1 comprising two shaft pairs, multiplying potentiometers having sliders coupled to the first shaft pair and receiving respectively poten' tials corresponding to said substitute variables, and another multiplying potentiometer coupled to the shaft of said shaft pair corresponding to one substitute variable and receiving a potential corresponding to another substitute variable, means for mixing the output potentials of the sliders of said multiplying potentiometers, and means under control or" the mixture for controlling the shafts of the second pair.

l4.- System according to claim 13 wherein the mixture takes place in accordance with a predetermined linear relation between substitute and derivative variables.

15. System according to claim 1 comprising means under control of said substitute variable to produce potentials corresponding to a sequence of derivative variables, said derivative variables multiplied with predetermined sequences of numerical coefiicients of P(z) forming algebraic sums, and thereby permanently representing said real and imaginary portions of P(z).

16. System according to claim 1 comprising a pair of primary control means, means under control of said primary means for producing adjustments roftsaid firstpair of adjustment means proportional to ,a-trigonom ltical function of two quantities represented by the positions of said primary control means; .and :means ,for controlling the second pair of adjustment means underi control of said primary control means-and-under control of the firstpair of adjustment means in accordance with the following trigonometrical formulae:

a and "b being the value of the two primary control positions.

17. System according'to claim 1 comprisingltw o shaft pairs andia pair ofiprimary shafts, .means iunderecontrol of said primary shafts for producing rotations of the first shaft pair roportional to a trignonometrical function of two quantities represented by the rotations :ofisaid primary shafts and means foricontrolling the .shafts of the second pair-under controLof vsaidaprimary 'shafts and under control of the shafts of the first pair in accordance with the following trigonometrical formulae:

x =sin a-sin bland y =cos (a+b)-cos (ab) a and b being the rotation values of the two primary shafts.

18. System according to claim 1 comprising'two shaft pairs corresponding to substitute and first derivative 'variables respectively and a pair of-potentiometers coupled to each of said shaftsreceiving reference potentials and potentials representing shaft variables respectively, and a third potentiometer coupled to one shaft of each pair receiving a potential difference representing the variable corresponding to another shaft, a number .of transfer stages for receiving the output potential from the sliders of said potentiometers so that one of said transferstages will deliver an output potential proportional to the difference of potentials delivered by thesliders of the potentiometers which receive the potential at terminals of the same denomination as their shaft; :while the other of said transfer stages will deliver an outputpotential proportional to the difference of potentials delivered by the slidersof the potentiometers which receive the potentials at terminals of the same denomination as their shafts; while the other of said transfer stages will deliver an output potential proportional to the potential delivered by the slider of the potentiometer which receives at its terminals the potential representing the variable corresponding to the shaft other than that coupled to its slider.

19. System according to claim 1 comprising a cathode ray tube having illuminating means and deflection assemblies receiving potentials representing substitute variables x xy means for rectifying and adding the output potentials representing said real and imaginary portions,

and means under control of the added potentials for controlling the illumination of said tubes so as to determine the roots of P(z).

20. System according to claim 1 comprising means for controlling the illumination under control of a potential of less than a predetermined value.

21. In a system for computing the real and imaginary portions of a polynomial P(z) of nth degree, two pairs'of shafts and means for driving the first shaft pair to angular positions corresponding to the values of substitute variables x and y representing the real-and imaginary portions of z, means under control of the first shaft pair for driving the second shaft pair to angular positions corresponding to the value of derivative variables 2:; and

y derived from said substitute variables including poten- I s mmin mp i ier a n mbe of assemblie .pszt n- 12 nme r sha in sl de coupl to sa d am n shst pair including an assembly of;potentio tnele, ,spreccivgingia reference potential difference and potential :difierences representing derivative variablesyx ,and 1 respectively, and assemblies of four potentiometers corresponding to the number of pairs of values of other derivative .variables x y derived from precedin g substitute variables, said four potentiometers having twosliders coupled toithe shaftcorresponding to y the two;pot entiometers qfleach shaft receiving potential diiferences representative of ederivative variables x,, and y and the two slide sof each shaft being coupled to different inputs of-said-amplifiers the outputs of which form potential differences representing derivative variables x and a four ntuitiplying potentiometer assemblies ,having sliders indicating in pairs the values of the coefficients of ,Pfiz) and-having resistances receiving impairs thepotentials representing all said pairs of derivative variables to be multiplied by said numerical coefiicients; whereby the multiplying potentiometers indicating coefiicientsra and 41 ha e resistances receiving a reference potential difference; one pair of further summing amplifiers having inputs receiving ,rcspectivelythe slider potentials of thosetmultiplying potentiometers indicating the-odd numbered coeificients of P(z) and having outputs forming of two algebraic sums 22 and 24; four further potentiometers havingJesistances receiving in pairs the outputs of said twofurther summing amplifiers and having slidersicoupled inpairs to the first-pairof shafts to deliver at the sliders qfisaid four -,fu;rther otentiometers the four potentials x 22, 1Z 12 and M 2; an 1 fina summing amp fiers rhaving inputs receiving respectively the potential differences-of those multiplying potentiometertsliders indicating the even numbered numerical :coeijicients of P(;) and delivering potential differences forming J94 get'ner the algebraic sums 21 and :23, plus'the potential differences of those further potentiometerrfiliders coupled to the shafts indicating values x and having resistances y; and receiving the potential diiferences representing sums E2 and Z4; and polarity inversion stages coupled in fronttof the resistances at least some of said potentiometer sliders coupled to said 'final surnming amplifier inputs sons to producettwo final summing amplifier-toutputs delivering respectively potential differences representing the values of the real portion a d im in ry p on 1 (z). -+a1Z -l-; ;1z o M2),-

22. System according to claim '21 *wherein the first alternatingcoefficients to'be entered are theoddt coefficients of P(z) and the second alternating coefficients to be entered are the even coefficients of =P( z) except coeificients a and a 23. System according to claim 21 wherein said pOlynomial is an algebraic polynomial with real numerical coefficients of a complex variable: z =x +jy written directly:

E4, the algebraic sum (a .y +a ly +a .y;+. derivative variables x and y, being related to substitute variables x1 and n by t e a q a and derivative variables of higher orders being related to first derivative variables x and y by relations (n being even).

24. System according to claim 21 comprising multiplying potentiometers coupled to the first shaft pair and receiving respectively potentials x and y and a multiplying potentiometer coupled to the shaft of said pair corresponding to variable y and receiving potentials x the output potentials of the sliders of said multiplying potentiometers being mixed in the input of the means controlling the shafts of the second pair; the mixtures taking place in accordance with a predetermined polynomial relation between variables x y and x .y

25; System according to claim 21 wherein for the production of the potentials 1x and $3 the shafts of the second pair support potentiometers receive potentials representing x and y and a third potentiometer is coupled to one of said shafts for receiving potentials representing the variable corresponding to the other shaft, there being further provided a number of transfer stages forreceiving the output potentials from the sliders of said potentiometers so that one of said transfer stages will deliver output potentials proportional to the difference of potentials delivered by the sliders of the potentiometers which receive the potentials at terminals of the same denomination as their shaft; while the other of said transfer stages will deliver output potentials proportional to the potentials delivered by the slider of the potentiometer which receives at its terminals the potentials representing the variable corresponding to the,

shaft other than that coupled to its slider.

26. System according to claim 21 comprising 'a pair of shafts of primary control, means for producing rotations proportional to a trigonometrical function of two quantities represented by the rotations of said primary control shafts, and including means for controlling the shafts of the second pair under control of said primary control shafts and under control of the shafts of the first pair in accordance with the following trigonometrical formulae:

x =sin asin b and y cos (a+b)cos (a-b) a and I) being the rotation values of the two primary control shafts.

27. System according to claim 21 comprising a cathode ray tube having an illumination control electrode and deflection assemblies and means for rectifying and adding the output potentials representing said real and imaginary portions, means under control of the added potentials for controlling theillumination of said tube so as to determine the roots of equation P(z)=0, and means controlling said electrode under control of the substitute variables and y 28. System according to claim 21 comprising means for controlling the illumination control electrode under control of potentials of less than a predetermined value.

29. In a system for computing, the real and imaginary portion of polynomial P(z) of the nth degree, at least two first adjustment means adjustable to positions corresponding to a pair of substitute variables, other adjustment means controlled by said first adjustment means to produce a first pair of derivative variables, a number of means corresponding to the number of other pairs of derivative variables, each for computing another pair of derivative variables, each receiving potentials representing the preceding pair of derivative variables and controlled by said other adjustment means respectively, means for multiplying potentials representing all said pairs of derivative variables with potentials representing all even and odd numbered numerical coefficients of P(z) respectively, to produce four series of product potentials of said derivative variables and said coefiicients, further means for summing the two series of product potentials derived from said odd numbered numerical coefiicients respectively, and means for multiplying 'each sum of said two series of product potentials with said substitute pair of variables to produce four product potentials of said sums and said substitute variables, and two final means for summing two of said last product potentials'and one sum of one of the other two series product potentials derived from said even numbered numerical coefficients to produce output potentials representing respectively the real and imaginary portions of P(z).

30. System according to claim 29 comprising a second pairof adjustment means and means under control of said first pair of adjustment means to control said second pair of adjustment means and to produce potentials representing respectively said first pair of derivative variables.

31. System according to claim 29 wherein said controlled means include sources of potentials adjusted by said first pair of adjustment means to produce said pair .of substitute variables and sources of potentials adjusted by said second pair of adjustment means under control of said substitute variables to produce said first pair of derivative variables.

32. System according to claim 29 wherein said computing means include a number of groups of sources of potentials corresponding to a number of'other pairs of derivative variables; each group including two sources of potentials representing the preceding pair of derivative variables and controlled by said second pair of adjustment means to produce potentials representing a succeeding pair of derivative variables.

33. System according to claim 29 wherein said multiplying means include four multiplying groups wherein two groups include a number oftsources of potentials representing a number of pairs of derivative variablesand adjusted to produce product potentials of derivative variables and the odd numbered coefficients of P(z).

34. System according to claim 29 wherein said further summing means and multiplying means include two means for adding product potentials derived respectively from a number of pairs of derivative variables and the odd numbered coeflicients of P(z) and two pairs of sources of potentials representing the sums of the last two product potentials and controlled by said first pair of adjustment means to produce four product potentials of sums of said last two products multiplied by pairs of substitute variables; and wherein said final summing means include two means for adding respectively two product potentials derived from said last pairs of sources and the productpotentials derived from the corresponding number of single derivative variables and the even numbered coefiicients of P(z) to produce two output potentials representing respectively the real and imaginary portions of P(z) p r V 35 In a system for computing the real and imaginary portions of a polynomial P(z) of nth degree, at least two adjustment means adjustable to positions corresponding to a pair of substitute variables representing the .real and imaginary portions of z, potential dividing means including sources of reference potentials received thereon and controlled by said adjustment means to reproduce potentials representing respectively said pair of substitute variables, other potential dividing means receiving said substitute variablesand controlled by said adjustment means to produce a first pair of derivative variables y derived from said substitute variables, a number of groups of further potential dividing means corresponding to a number of other pairs of derivative variables each particular pair of said other pairs of derivative.

variables being derived from derivative variables preceding said particular pair; each group of said further poten- .duec mtent a s r p sentin a eedi pa o d r vative variables; a number of potential dividing meansreceiving all said pairs of derivative Variables and controlled ,by all even numbered numerical coeflicients of Hz) ,to produce two series of product potentials of said derivative variables ,with said even numerical coefficients of ,P. (z) ,a ,number of further potential dividing means receivingall said pairs of derivative variables and controlled ,by the odd numbered numerical coefficients of P(z) to produce further series of product potentials of said derivative variables of said odd numbered numericalpoefiicientssof P(z) means for summing said further two series of product potentialsrespectively, pairs of potential dividing means receiving the sumsof said further two series of product potentials and controlled by potentials representing said substitute variables to produce four product potentials of sums of said further two series of products multiplied 1by pairs of substitute variables; and ,finahrneans for summing two of thetlast product potentialsandonesuni of preductpotentialsderived from one of said first two series of product potentials to produce output potentials representing respectively the real and imaginary portions of"P(z).

3 6. System according to claim 35 comprising-two pairs of shafts and means for,driving the first shaft pair to angular positions corresponding to the values of substitute variables x and y means undercontrol of-thefirst shaft pair for driving-the second shaft pairto angular positions corresponding to the value of variables x and y; including several potential dividing means 'supplied by reference potential difference and a potential difference representing variables x and y, respectively, and having dividers coupled to said first shaft pair to control said second shaft pair and to deliver potential differences representing the values of products of angular positions x y respectively.

'37. System according to claim 35 comprising polarity inversion means connected to at least some points of said potentials and to at least some of said summing means.

'38. System accordingto claim 35, comprising a num- .ber of summing amplifiers and a number of assemblies. of potential dividing means having dividers coupled to said .second shaft pair receiving areference potential difference ,and,a potential dilferencerepresenting derivative variables x, and y; respectively; and assemblies of four potential d idi mean o sp ndin to h n m e o o h ,nai pf va ue o de at v v i es nan a. ai u p tentialdivi ding means having two dividers coupled to theshafheorresponding to x and two other dividersegupledto the shaft corresponding toy two potential dividing means ,of each shaft receiving potential ditferences representative ,of derivative variables x,, and y and the two dividers of each shaft representing products of variables x y and x,, y,, being coupled to different inputs of said further amplifiers the ,outputsof which form potential differences representing variables @x andy a 39. System according to claim 35 wherein said multiplying means ,include four multiplying potential dividing 16 a semblie hav n d de s i di ti t values f the coeflicients of four ,algebraic sums 21 to 24; said potential dividing means receiving respectively the potentials to be multiplied by said numerical coefiicients; the multiplying potential dividing means including co eff cients n and z receiving a reference potential difference; two gf urther summing amplifiers having inputs .rcceiviugres pectively the divider potentials of those multiplyingpotential dividingmeans indicating the coefficients entering in .the forrnation of algebraic sums E2 and 24-; ,four further potential dividing means having dividers coupled ,in .pairs to the first pair of shafts corresponding to variables x; and 31 said two further summing ampli- ,fiers,having,outputssupplying each two of said latter potential dividing means, thereby to deliver at the dividers .of saidiour furtherpotential dividing means the four potentials 14 22-2124, x 224 and y 222l, 212, L 23, 24,, being; definedasiollows:

40. S ystemaccording to claim 35 wherein said final summing means include-two summing amplifiers having .inputs receiving ,respectively the potential difference of .those further ,potentiahdividers indicating the numerical coefficients entering gin the, formation of the algebraic .sum .21 ;the potential difference of those further potential dividers coupled to ,the shafts indicating values ,x and y and supplying the potential difference repre- ,senting sums EZ and 224; the potential difference of those ,furtherpotentiahdividers indicating the numerical co- .etficients entering .in the formation of algebraic ,E2; and ,the potentialsofthese further potential dividers coupled -to the shafts indicating values x and M and supplying the potential difference representing sums 2:4 and E2 521, -2.2, Z3, 24,.b eing-defined as follows:

respeet ively :potential differences representing the values ,pf the real portiou Rflz) and :imaginary portion I(z) :of the ,eguation .having .said coefficients.

References Cited in the file of this patent -,UNIIED ;.STATES JPATENTS 2,444,549 Anderson July 6, 1948 2,454;549 Brown'et al. Nov. 25, 1948 2,538,253 Lakatos et al. Jan. 16, 1951 2,558,430 Goldberg June 26, 1951 2,613,032 Serellet al. Oct. 7, 1952 

